GEOMETRY CHAPTER 3 Parallel and Perpendicular Lines Section

GEOMETRY CHAPTER 3 Parallel and Perpendicular Lines

Section 3 -1: Relationships between Lines � Parallel lines: two lines that lie on the same plane and do not intersect � Perpendicular lines: two lines that intersect to form a right angle � Skew lines: two lines that do not lie in the same plane. They never intersect

� Parallel planes: two planes that do not intersect � Line perpendicular to a plane: a line that intersects a plane in a point and that is perpendicular to every line in the plane that intersects it

Section 3 -2: Theorems about Perpendicular Lines � B A 1 m 2 4 3 n

� B A 1 D 2 C F G 3 E 4 H

Section 3 -3: Angles formed by Transversals � � Transversal: a line that intersects two or more coplanar lines at different points. Corresponding angles: occupy the same position on different lines (i. e. top left) � 1 and 5 � 2 and 6 � 3 and 7 � 4 and 8 � 1 2 3 4 5 6 7 8 Alternate Interior angles: lie between the two t lines on opposite sides of the transversal � 3 and 6 � 4 and 5 1 2 34 5 6 7 8

t Alternate Exterior angles: lie outside the two lines on opposite sides of the 1 2 3 4 transversal � � 1 and 8 � 2 and 7 5 6 7 8 t Same-side Interior angles: lie between 1 2 the two lines on the same side 3 4 of the transversal 5 6 � � 3 and 5 � 4 and 6 7 8

Section 3 -4: Parallel Lines and Transversals � Postulate 8: Corresponding Angles Postulate � If two parallel lines are cut by a transversal, then corresponding angles are congruent. � Theorem 3. 5: Alternate Interior Angles Theorem � If two parallel lines are cut by a transversal, then alternate interior angles are congruent. � Theorem 3. 6: Alternate Exterior Angles Theorem � If two parallel lines are cut by a transversal, then alternate exterior angles are congruent. � Theorem 3. 7: Same-Side Interior Angles Theorem � If two parallel lines are cut by a transversal, then sameside interior angles are supplementary. (add up to 180 )

Section 3 -5: Showing Lines are Parallel � The converse of an if-then statement is the statement formed by switching the hypothesis and the conclusion. � Example: Statement- If you live in Sacramento, then you live in California. Converse- If you live in California, then you live in Sacramento.

� � Postulate 9: Corresponding Angles Converse: � If 1 5, then r s. Theorem 3. 8: Alternate Interior Angles Converse: � If 4 5, then r s. Theorem 3. 9: Alternate Exterior Angles Converse: � If 1 8, then r s. Theorem 3. 10: Same-Side Interior Angles Converse: � If m 3 + m 5 =180 , then r s. 1 3 4 5 8 s r

Section 3 -6: Using Perpendicular & Parallel Lines � Postulate 10: Parallel Postulate � If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. P m l � m Postulate 11: Perpendicular Postulate � If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line. P l

� Theorem 3. 11 � If two lines are parallel to the same line, then they are parallel to each other. q � s r Theorem 3. 12 � In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. m n p

6 Ways to Show that 2 lines are Parallel 1. 2. 3. 4. 5. 6. Corresponding Angles Converse Alternate Interior Angles Converse Alternate Exterior Angles Converse Same-Side Interior Angles Converse Theorem 3. 11 (from this section) Theorem 3. 12 (from this section)

Section 3 -7: Tranformations � � � Transformation: an operation that maps or moves a figure onto an image Image: new figure after a transformation Translation: also known as a slide. Image remains unchanged but moved to a different position. Reflection: Also know as a flip. The new figure is the mirror image of the original over a given line. Rotation: also known as a turn. The figure is turned about a given point.